🧬 biology

What a size-only threshold for holding cancer treatment loses in a spatial tumour model

This study demonstrates that the size-only treatment hold threshold derived from non-spatial ODE models fails to prevent premature progression in spatial agent-based models, and while incorporating measurement history and tumor geometry offers marginal improvements in drug-free time, these gains are inconsistent and fail to reliably meet safety tolerances across all testing intervals.

Original authors: Arjun Grandhe

Published 2026-10-30

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Arjun Grandhe

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Cancer treatment often relies on a delicate balance: using drugs to shrink a tumor while trying to avoid wiping out the body's own defenses or allowing the cancer to become permanently resistant. One promising approach, known as adaptive therapy, treats the cancer only enough to keep it in check, then pauses the medication. The goal is to let the remaining, drug-sensitive cancer cells survive and compete with the drug-resistant ones, effectively keeping the resistant population in line. However, doctors cannot monitor a tumor every single day; they must wait for scheduled visits. This creates a critical gap: if a doctor stops treatment based on a tumor's current size, how can they be sure the tumor won't grow too large and become dangerous before the next appointment?

For years, scientists have used mathematical models to calculate a "safe size" threshold. These models suggest that if a tumor is smaller than a specific number of cells when treatment stops, it will not grow big enough to cause harm by the time the patient returns for their next check-up. This calculation works well in simple, theoretical models where every cell is assumed to mix perfectly with every other cell, like sugar dissolving in a cup of tea. But real tumors are not well-mixed soups; they are complex, three-dimensional structures where cells are packed together, and drugs must physically diffuse through the tissue to reach them. A new study by Arjun Grandhe investigates whether these simple, size-based rules hold up when applied to a more realistic, spatial model of a tumor.

Grandhe set up a computer simulation that mimics a tumor as a crowded city of individual cells on a grid. In this digital world, cells can only divide if they have an empty space next to them, and the drug must travel through the grid to reach them, just as it would in a human body. The simulation included two types of cells: those that die when hit by the drug and those that are resistant to it. The researcher then tested the standard "safe size" rule derived from the simple models against this complex, spatial simulation. The results were stark. At short intervals between check-ups, such as one or two weeks, the standard rule failed dramatically. When the tumor was allowed to grow from the size predicted to be safe, it exploded past the danger limit in nearly every single simulation run. In fact, at a one-week interval, the tumor progressed prematurely in 97.6% of the cases, meaning the rule was dangerously wrong. However, the study found that at longer intervals of 42 to 84 days, the derived threshold was actually safe, as the construction's own margin of safety outweighed the shortfall.

The study explored why this happened. It turned out that the simple models, which assume a well-mixed population, could not account for the physical constraints of a real tumor. In the complex simulation, the tumor's growth speed and behavior depended heavily on its shape and how the cells were arranged, not just on how many cells there were. Even when the researcher adjusted the rules to match the specific growth speed of the simulated tumor, the simple size threshold remained unsafe. The gap between the simple prediction and the complex reality was so large that it could not be explained by a simple difference in growth rates alone. The failure appeared to stem from the moment the model switched from a theoretical, mixed population to a spatial one where cells are locked in place and drugs move slowly.

The research then asked if there was a better way to predict danger. Instead of looking only at the current size of the tumor, the study tested whether knowing the tumor's history—how fast it had grown since the last visit, how long it had been treated, and its shape—could improve the prediction. In the computer simulation, using this extra information was a game-changer. When the tumor size was measured perfectly, knowing the history of the tumor's growth allowed the model to predict a dangerous progression with much higher accuracy than size alone. The shape of the tumor also provided valuable clues. However, this advantage was fragile. When the researchers added a realistic amount of measurement error, simulating the imperfections of real-world medical imaging, the benefit of using history and shape almost vanished. The extra information was only useful if the measurements were perfect, which is rarely the case in a clinic.

Finally, the study tested whether using this extra information to make treatment decisions would actually help patients. The researchers created a new rule that held off treatment not just when the tumor was small, but also when its history and shape suggested it was safe to wait. Compared to the standard rule that only looked at size, this new approach slightly reduced the amount of time patients spent on drugs and slightly delayed the time until the tumor grew too large. The improvement was small and inconsistent, resting on two of the four intervals tested: it showed a gain at 7 and 28 days, but did not show a clear benefit at 14 or 56 days. More importantly, the new rule did not consistently prove itself to be safer than the old one; in several tests, the risk of the tumor growing too large remained too high to be considered a reliable improvement.

The study concludes that while simple size thresholds are easy to use, they can be dangerously misleading in the complex, spatial reality of a tumor, especially when check-ups are frequent. The extra information found in a tumor's history and shape does contain valuable clues about its future behavior, but capturing that information requires measurements so precise that they are currently beyond the reach of standard medical practice. Until we can measure tumors with perfect accuracy, the promise of using complex data to improve adaptive therapy remains largely theoretical. The findings suggest that the path to better cancer control lies not just in smarter rules, but in understanding the physical limits of how we can see and measure the disease.

Technical Summary: What a size-only threshold for holding cancer treatment loses in a spatial tumour model

Problem Statement
Adaptive therapy aims to control tumour growth by maintaining drug-sensitive cells to compete with resistant ones, often employing a "hold" strategy where treatment is paused while the tumour is small and resumed upon regrowth. In clinical practice, decisions are made at discrete intervals based on tumour size measurements. Gallagher et al. (2025) derived a size-only threshold ( $N^*$ ) for holding treatment in non-spatial ordinary differential equation (ODE) models, ensuring that a tumour starting below this size would not reach a progression limit within the monitoring interval.

This study investigates whether this derived threshold remains safe when applied to a spatial agent-based model (ABM) with discrete monitoring. Specifically, it asks:

Does the ODE-derived threshold maintain a stated tolerance for "premature progression" (progression occurring during a treatment hold) in a spatial ABM?
If not, where in the modeling chain (ODE $\to$ stochastic CTMC $\to$ spatial ABM) does the safety gap appear?
Does incorporating information beyond size (measurement history, tumour geometry) improve the prediction of premature progression and lead to better treatment control under a fixed risk constraint?

Methodology
The study utilized a spatial agent-based model (ABM) on a $208 \times 208$ lattice containing sensitive and resistant cells. Drug concentration followed a reaction-diffusion equation. Treatment was governed by a discrete-monitoring policy with intervals ( $\tau$ ) of 7, 14, 28, 42, 56, and 84 days.

Threshold Derivation: The study used the Gallagher et al. (2025) construction to calculate $N^*(\tau)$ , the size from which an untreated ODE trajectory reaches the progression limit in exactly time $\tau$ .
Model Chain Analysis: To locate the source of safety failures, the study compared the ABM against a chain of simpler models: a deterministic ODE, a well-mixed continuous-time Markov chain (CTMC-nominal), and a CTMC with pharmacokinetic dynamics (CTMC-pk).
Safety Metrics: The primary endpoint was the fraction of runs with premature progression ( $p_{prem}$ ). A "safe threshold" ( $N_{safe}$ ) was defined as the largest threshold keeping $p_{prem} \le 0.05$ .
Recalibration: The study tested if the safety gap persisted when the growth law and threshold were recalibrated to the ABM's own drug-free regrowth dynamics.
Prediction and Policy: Logistic regression models were trained to predict premature progression using: (1) size only, (2) size + measurement history, (3) size + history + state (drug/resistant count), and (4) size + history + state + geometry. These models were then used to drive "hold" rules, which were tuned to the same nominal risk constraint ( $p_{prem} \le 0.05$ ) as the size-only rule to compare outcomes (on-drug fraction and restricted mean time to progression, RMST).

Key Results

Failure of the Derived Threshold in Spatial Models:
- At short monitoring intervals (7, 14, and 28 days), the ODE-derived threshold was unsafe in the ABM. Premature progression occurred in 97.6%, 97.4%, and 48.0% of runs, respectively, far exceeding the 5% tolerance.
- At longer intervals (42–84 days), the threshold became safe, but this was due to the ODE construction's inherent "slack" (margin of safety) outweighing the model transfer error, not because the ODE accurately described the ABM.
- Decomposition: The safety gap was decomposed into a "derivation term" (ODE construction vs. ODE safe threshold) and a "transfer term" (ODE vs. ABM). The transfer term was consistently negative (unsafe), ranging from -4.40% to -8.95% at 7–56 days. Most of this gap (84.2% to 94.1% at 7–28 days) appeared in the step from the well-mixed stochastic model (CTMC) to the spatial ABM. This step bundles contact-limited division, lattice discreteness, and drug transport, so the specific mechanism remains unidentified.
- Recalibration Failure: Recalibrating the threshold to the ABM's own drug-free regrowth (even using the 5th percentile of regrowth times) did not eliminate the safety shortfall. The ABM safe threshold remained significantly lower than the recalibrated threshold, suggesting that tumour size alone is insufficient to predict near-term regrowth after treatment history.
Predictive Value of History and Geometry:
- Prediction: With error-free measurements, adding measurement history to size reduced the test log-loss for predicting premature progression by 48.6% (pooled). Adding tumour geometry (rim cells, radius of gyration, etc.) provided a further 30.5% reduction.
- Measurement Noise: This advantage was highly sensitive to measurement error. With 10% noise in size readings, the history model was only 2.0% better than a size-only model with exact readings. Most of the predictive gain depended on error-free data.
- Within-Size Prediction: History and geometry improved predictions even when tumours were of the same size, indicating that size is not a sufficient state for determining regrowth speed in this spatial model.
Policy Performance:
- A hold rule using history and image geometry (primary rule) was compared against a size-only rule, both tuned to the same nominal risk.
- Outcomes: The history-and-image rule reduced the on-drug fraction by 1.4 percentage points and increased RMST by 4.0 days over 750 days.
- Robustness: These gains were fragile. They rested on only two of four intervals (7 and 28 days). At 7 days, the rule's risk sat exactly on the non-inferiority margin; at 14 days, non-inferiority was not shown. Over a 365-day horizon, the gain in time disappeared.
- Risk Calibration: The models' risk estimates were poorly calibrated when used to drive the rules (observed-to-expected ratios ranged from 0.28 to 2.46), meaning the rules were tuned via simulation outcomes rather than model probability. On new runs, the history-and-image rule's risk was not statistically shown to meet the tolerance at any interval.

Significance and Claims
The paper claims that a size-only threshold derived from non-spatial ODE models does not transfer safely to a spatial agent-based model at clinically relevant short monitoring intervals. The safety shortfall persists even after recalibrating for regrowth speed, suggesting that the "state" of a tumour in a spatial model requires more than just its current size; treatment history and spatial configuration matter.

While measurement history and geometry significantly improve the prediction of premature progression in the simulator, this predictive advantage does not translate into a robust improvement in control (reduced drug exposure or delayed progression) under the tested one-step hold rules and risk constraints. The study concludes that better prediction did not produce a robust improvement in control in this specific model setting, and the observed gains were conditional on specific modeling choices (cap, horizon, non-inferiority margin) and fragile to statistical variation. The practical value of these features remains untested with real-world measurement noise.

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